A retrospective on the accuracy and precision of plotless forest density estimators in ecological studies

Sampling point-to-tree distances is a simple plotless technique for estimating forest density that is readily applied in modern stands and retroactively with historical surveys. Although plotless density estimators (PDEs) have been applied in over 1000 ecological publications, the accuracy and precision of the techniques remain poorly understood and depend on the statistical estimator used, the underlying spatial pattern of the forest sampled, and the tree survey methodology. The four most commonly applied PDEs are related formulations: Cottam, Pollard, Morisita, and Shanks, a family of equations that differ in the order of mathematical operations. Since the 1950s, the Cottam IV PDE has found common use as the "point-quarter method." The Pollard PDE prevails in the statistical literature. Both Cottam and Pollard PDEs are theoretically rigorous for trees distributed according to a complete spatial randomness (CSR) spatial point process. The Morisita PDE was developed in a 1957 publication, with four-tree (Morisita IV) and two-tree (Morisita II) variants, and is the basis for higher distance rank g-tree estimators. The Shanks PDE is formally described here for the first time. We review and evaluate the performance of these four PDEs on CSR and a variety of non-CSR forests using spatial patterns simulated from known spatial point processes, 14 mapped modern stands, and historical public land surveys (PLSs). We found that the Cottam and Pollard PDEs lacked accuracy for non-CSR patterns. The Morisita PDEs are appropriate for non-CSR forests, but the Morisita IV has sensitivity to local dispersion. The Morisita II PDE has high accuracy even under non-CSR distributions yielding density estimates within 10% of the true value for a variety of non-CSR patterns, but has considerable variability at small sample sizes. In conjunction with the Morisita II, the potentially biased Cottam and Pollard PDEs can be indicators of the type of non-CSR pattern. No plotless estimator is efficacious for use with small sample sizes such as found in a single stand. Morisita II PDE is recommended as a robust choice for sampling for large and non-CSR data sets such as the PLS witness tree database.